Gaussian Approximations of Small Noise Diffusions in Kullback-Leibler Divergence

TL;DR

This paper develops and analyzes Gaussian approximation methods for small noise diffusions using Kullback-Leibler divergence, providing theoretical validation and practical guidance for their application.

Contribution

It introduces simple ODE-based Gaussian approximations for diffusions and demonstrates their accuracy in the small noise regime, supported by theoretical analysis.

Findings

Gaussian approximations are accurate in small noise regimes

Explicit ODEs for mean and covariance simplify computations

Results support Gaussian processes in diffusion approximation

Abstract

We study Gaussian approximations to the distribution of a diffusion. The approximations are easy to compute: they are defined by two simple ordinary differential equations for the mean and the covariance. Time correlations can also be computed via solution of a linear stochastic differential equation. We show, using the Kullback-Leibler divergence, that the approximations are accurate in the small noise regime. An analogous discrete time setting is also studied. The results provide both theoretical support for the use of Gaussian processes in the approximation of diffusions, and methodological guidance in the construction of Gaussian approximations in applications.